I consider the Dirichlet problem $$ \begin{cases} \Delta u(x) = 0, \quad x \in D,\\ u|_{\partial D} = \varphi, \end{cases} $$ where $\varphi \in C^{1,\alpha}(\partial D)$ and $D$ is a bounded domain in $\mathbb R^n$, $n \geq 2$, with $\partial D \in C^2$. There exists the unique classical solution $u$ to this Dirichlet problem and it belongs to $C^\infty(D) \cap C^1(\overline D)$ (the fact that $u \in C^1(\overline D)$ follows, for example, from Theorem 8.2.15 of W. Hackbusch, "Integral equations", 1995).
But is it true that the solution operator is a continuous operator from $C^{1,\alpha}(\partial D)$ to $C^1(\overline D)$?